Constructions of asymptotically shortest k-radius sequences

Abstract

Let k be a positive integer. A sequence s over an n-element alphabet A is called a k-radius sequence if every two symbols from A occur in s at distance of at most k. Let f_k(n) denote the length of a shortest k-radius sequence over A. We provide constructions demonstrating that (1) for every fixed k and for every fixed e>0, f_k(n) = n^2/(2k) +O(n^(1+e)) and (2) for every k, where k is the integer part of n^a for some fixed real a such that 0 < a <1, f_k(n) = n^2/(2k) +O(n^b), for some b <2-a. Since f_k(n) >= n^2/(2k) - n/(2k), the constructions give asymptotically optimal k-radius sequences. Finally, (3) we construct optimal 2-radius sequences for a 2p-element alphabet, where p is a prime.